#8455692

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,6,5)(3,8,4,7)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,22)(23,24)(26,30)(31,37)(32,38)(39,40);(1,3,6,4)(2,7,5,8)(9,11,15,13)(10,16,20,17)(12,23,19,21)(14,22,18,24)(26,30)(27,33)(32,38)(34,35);(9,11,15,13)(10,17,20,16)(12,21,19,23)(14,22,18,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(2,5)(7,8)(9,12,15,19)(10,18,20,14)(11,21,13,23)(16,22,17,24)(25,27,29,33)(26,31,30,37)(28,35,36,34)(32,40,38,39);(1,4,6,3)(2,8,5,7)(9,13,15,11)(10,16,20,17)(12,23,19,21)(14,24,18,22);(1,5,6,2)(3,8,4,7)(9,14,15,18)(10,19,20,12)(11,22,13,24)(16,21,17,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362888);